Semantic Scholar€¦ · CAHN-HILLIARD INPAINTING AND A GENERALIZATION FOR GRAYVALUE IMAGES MARTIN BURGER∗, LIN HE†, AND CAROLA-BIBIANE SCHONLIEB¨ ‡ Abstract. The Cahn-Hilliard

CAHN-HILLIARD INPAINTING AND A GENERALIZATION FORGRAYVALUE IMAGES

MARTIN BURGER∗, LIN HE† , AND CAROLA-BIBIANE SCHONLIEB‡

Abstract. The Cahn-Hilliard equation is a nonlinear fourth order diffusion equation originating in materialscience for modeling phase separation and phase coarsening in binary alloys. The inpainting of binary images usingthe Cahn-Hilliard equation is a new approach in image processing. In this paper we discuss the stationary stateof the proposed model and introduce a generalization for grayvalue images of bounded variation. This is realizedby using subgradients of the total variation functional within the flow, which leads to structure inpainting withsmooth curvature of level sets.

Key words. Cahn-Hilliard equation, TV minimization, image inpainting

AMS subject classifications. 49J40

1. Introduction. An important task in image processing is the process of filling in missingparts of damaged images based on the information obtained from the surrounding areas. It isessentially a type of interpolation and is referred to as inpainting. Given an image f in a suitableBanach space of functions defined on Ω ⊂ R2, an open and bounded domain, the problem is toreconstruct the original image u in the damaged domain D ⊂ Ω, called inpainting domain. Inthe following we are especially interested in so called non-texture inpainting, i.e., the inpaintingof structures, like edges and uniformly colored areas in the image, rather than texture.In the pioneering works of Caselles et al. [14] (with the term disocclusion instead of inpainting)and Bertalmio et al. [5] partial differential equations have been first proposed for digital non-texture inpainting. The inpainting algorithm in [5] extends the graylevels at the boundary of thedamaged domain continuously in the direction of the isophote lines to the interior. The resultingscheme is a discrete model based on the nonlinear partial differential equation

ut = ∇⊥u · ∇∆u,

to be solved inside the inpainting domain D using image information from a small strip around D.The operator ∇⊥ denotes the perpendicular gradient (−∂y, ∂x). To avoid the crossing of level linesinside the inpainting domain, intermediate steps of anisotropic diffusion are implemented into themodel.In subsequent works variational models, originally derived for the tasks of image denoising, de-blurring and segmentation, have been adopted to inpainting. In contrast to former approaches(like [5]) the proposed variational algorithms are applied to the image on the whole domain Ω.This procedure has the advantage that inpainting can be carried out for several damaged domainsin the image simultaneously and that possible noise outside the inpainting domain is removed atthe same time. The general form of such a variational inpainting approach is

u(x) = argminu∈H1

(J(u) = R(u) +

12‖λ(f(x)− u(x))‖2H2

),

where f ∈ H2 (or f ∈ H1 depending on the approach) is the given damaged image and u ∈ H1

is the restored image. H1,H2 are Banach spaces on Ω and R(u) is the so called regularizingterm R : H1 → R. The function λ is the characteristic function of Ω \D multiplied by a (large)constant, i.e., λ(x) = λ0 >> 1 in Ω \D and 0 in D. Depending on the choice of the regularizing

∗Institut fur Numerische und Angewandte Mathematik, Fachbereich Mathematik und Informatik,Westfalische Wilhelms Universitat (WWU) Munster, Einsteinstrasse 62, D 48149 Munster, Germany. Email:[email protected]

†Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences,Altenbergerstraße 69 A-4040, Linz, Austria. Email: [email protected]

‡DAMTP, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom. Email:[email protected]

1

2 M. Burger, L. He, and C.-B. Schonlieb

term R(u) and the Banach spaces H1, H2 various approaches have been developed. The mostfamous model is the total variation (TV) model, where R(u) = |Du| (Ω), is the total variation ofu, H1 = BV (Ω) the space of functions of bounded variation (see Appendix A for the definition offunctions of bounded variation) and H2 = L2(Ω), cf. [19, 17, 34, 35]. A variational model with

a regularizing term of higher order derivatives, i.e., R(u) =∫Ω(1 +

(∇ · ( ∇u

|∇u| ))2

)|∇u| dx, is theEuler elastica model [16, 28]. Other examples are the active contour model based on the Mumfordand Shah segmentation [36], and the inpainting scheme based on the Mumford-Shah-Euler imagemodel [22].Second order variational inpainting methods (where the order of the method is determined by thederivatives of highest order in the corresponding Euler-Lagrange equation), like TV inpainting(cf. [35], [17], [18]) have drawbacks as in the connection of edges over large distances or thesmooth propagation of level lines (sets of image points with constant grayvalue) into the damageddomain. In an attempt to solve both the connectivity principle and the so called staircasing effectresulting from second order image diffusions, a number of third and fourth order diffusions havebeen suggested for image inpainting.A variational third order approach to image inpainting is the CDD (Curvature Driven Diffusion)method [18]. While realizing the Connectivity Principle in visual perception, (i.e., level lines areconnected also across large inpainting domains) the level lines are still interpolated linearly (whichmay result in corners in the level lines along the boundary of the inpainting domain). This hasdriven Chan, Kang and Shen [16] to a reinvestigation of the earlier proposal of Masnou and Morel[28] on image interpolation based on Euler´s elastica energy. In their work the authors present thefourth order elastica inpainting PDE which combines CDD and the transport process of Bertalmioet. al [5]. The level lines are connected by minimizing the integral over their length and theirsquared curvature within the inpainting domain. This leads to a smooth connection of level linesalso over large distances. This can also be interpreted via a second boundary condition, necessaryfor an equation of fourth order. Not only the grayvalues of the image are specified on the boundaryof the inpainting domain but also the gradient of the image function, namely the directions of thelevel lines are given. Further, also combinations of second and higher order methods exist, e.g.[27]. The combined technique is able to preserve edges due to the second order part and at thesame time avoids the staircasing effect in smooth regions. A weighting function is used for thiscombination.The main challenge in inpainting with higher order flows is to find simple but effective modelsand to propose stable and fast discrete schemes to solve them numerically. To do so also themathematical analysis of these approaches is an important point, telling us about solvability andconvergence of the corresponding equations. This analysis can be very hard because often theseequations do not admit a maximum or comparison principle and sometimes do not even have avariational formulation.A new approach in the class of fourth order inpainting algorithms is inpainting of binary imagesusing a modified Cahn-Hilliard equation, as proposed in [7], [8] by Bertozzi, Esedoglu and Gillette.The inpainted version u of f ∈ L2(Ω) assumed with any (trivial) extension to the inpainting domainis constructed by following the evolution of

ut = ∆(−ε∆u+1εF ′(u)) + λ(f − u) in Ω, (1.1)

where F (u) is a so called double-well potential, e.g., F (u) = u2(u− 1)2, and - as before:

λ(x) =

λ0 Ω \D0 D

is the characteristic function of Ω \ D multiplied by a constant λ0 >> 1. The Cahn-Hilliardequation is a relatively simple fourth-order PDE used for this task rather than more complexmodels involving curvature terms. Still the Cahn-Hilliard inpainting approach has many of thedesirable properties of curvature based inpainting models such as the smooth continuation of level

Cahn-Hilliard and BV-H−1 inpainting 3

lines into the missing domain. In fact in [8] the authors prove that in the limit λ0 →∞ a stationarysolution of (1.1) solves

∆(ε∆u− 1εF

′(u)) = 0 in Du = f on ∂D

∇u = ∇f on ∂D,(1.2)

for f regular enough (f ∈ C2). Moreover its numerical solution was shown to be of at least anorder of magnitude faster than competing inpainting models, cf. [7].

In [8] the authors prove global existence and uniqueness of a weak solution of (1.1). Moreover,they derive properties of possible stationary solutions in the limit λ0 → ∞. Nevertheless theexistence of a solution of the stationary equation

∆(−ε∆u+1εF ′(u)) + λ(f − u) = 0 in Ω, (1.3)

remains unaddressed. The difficulty in dealing with the stationary equation is the lack of anenergy functional for (1.1), i.e., the modified Cahn-Hilliard equation (1.1) cannot be representedby a gradient flow of an energy functional over a certain Banach space. This is because the fidelityterm λ(f − u) isnt symmetric with respect to the H−1 inner product. In fact the most evidentvariational approach would be to minimize the functional∫

Ω

(ε

2|∇u|2 +

1εF (u)) dx+

12‖λ(u− f)‖2−1 . (1.4)

This minimization problem exhibits the optimality condition

0 = −ε∆u+1εF ′(u) + λ∆−1 (λ(u− f)) ,

which splits into

0 = −ε∆u+ 1εF

′(u) in D0 = −ε∆u+ 1

εF′(u) + λ2

0∆−1 (u− f) in Ω \D.

Hence the minimization of (1.4) translates into a second order diffusion inside the inpaintingdomain D, whereas a stationary solution of (1.1) fulfills

0 = ∆(−ε∆u+ 1εF

′(u)) in D0 = ∆(−ε∆u+ 1

εF′(u)) + λ0 (f − u) in Ω \D.

One challenge of this paper is to extend the analysis from [8] by partial answers to ques-tions concerning the stationary equation (1.3) using alternative methods, namely by fixed pointarguments. We shall prove

Theorem 1.1. Equation (1.3) admits a weak solution in H1(Ω) provided λ0 ≥ C 1ε3 , for a

positive constant C depending on |Ω|, |D|, and F only.We will see in our numerical examples that the condition λ0 ≥ C 1

ε3 in Theorem 1.1 is naturallyfulfilled, since in order to obtain good visual results in inpainting approaches λ0 has to be chosenrather large in general. Note that the same condition also appears in [8] where it is needed toprove the global existence of solutions of (1.1).

The second goal of this paper is to generalize the Cahn-Hilliard inpainting approach to gray-value images. This is realized by using subgradients of the TV functional within the flow, whichleads to structure inpainting with smooth curvature of level sets. We motivate this new approachby a Γ−convergence result for the Cahn-Hilliard energy. In fact we prove that the sequence offunctionals for an appropriate time-discrete Cahn-Hilliard inpainting approach Γ-converges to afunctional regularized with the total variation for binary arguments u = χE , where E is someBorel measurable subset of Ω. This is stated in more detail in the following Theorem.


Theorem 1.2. Let f, v ∈ L2(Ω) be two given functions, and τ > 0 a positive parameter. Letfurther ‖·‖−1 be the norm in H−1(Ω), defined in more detail in the Notation section, and |Du| (Ω)denote the total variation of the distributional derivative Du (cf. Appendix B). Then the sequenceof functionals

Jε(u, v) =∫

Ω

(ε

2|∇u|2 +

1εF (u)

)dx+

12τ

‖u− v‖2−1 +λ0

2

∥∥∥∥u− λ

λ0f −

(1− λ

λ0

)v

∥∥∥∥2

−1

Γ−converges for ε→ 0 in the topology of L1(Ω) to

J(u, v) = TV (u) +12τ

‖u− v‖2−1 +λ0

2

∥∥∥∥u− λ

λ0f −

(1− λ

λ0

)v

∥∥∥∥2

−1

,

where

TV (u) =

C0 |Du| (Ω) if u = χE for some Borel measurable subset E ⊂ Ω+∞ otherwise,

with χE denotes the characteristic function of E and C0 = 2∫ 1

0

√F (s) ds.

Remark 1.3. Setting v = uk and a minimizer u of the functional Jε(u, v) to be u = uk+1,the minimization of Jε can be seen as an iterative approach with stepsize τ to solve (1.1).

Now, by extending the validity of the total variation functional TV (u) from functions u = 0 or1 to functions |u| ≤ 1 we receive an inpainting approach for grayvalue images rather than binaryimages. We shall call this new inpainting approach TV − H−1 inpainting and define it in thefollowing way: The inpainted image u of f ∈ L2(Ω), shall evolve via

ut = ∆p+ λ(f − u), p ∈ ∂TV (u), (1.5)

with

TV (u) =

|Du| (Ω) if |u(x)| ≤ 1 a.e. in Ω+∞ otherwise.

(1.6)

The inpainting domain D and the characteristic function λ(x) are defined as before for the Cahn-Hilliard inpainting approach. The space BV (Ω) is the space of functions of bounded variationon Ω (cf. Appendix B). Further ∂TV (u) denotes the subdifferential of the functional TV (u) (cf.Appendix B for the definition). The L∞ bound in the definition of the TV functional (1.6) is quitenatural as we are only considering digital images u whose grayvalue can be scaled to [−1, 1]. It isfurther motivated by the Γ− convergence result of Theorem 1.2.

A similar form of the TV-H−1 inpainting approach already appeared in the context of de-composition and restoration of grayvalue images, see for example [38], [33], and [26]. Further, inBertalmio at al. [6] an application of the model from [38] to image inpainting has been proposed.In contrast to the inpainting approach (1.5) the authors in [6] only use a different form of the TV-H−1 approach for a decomposition of the image into cartoon and texture prior to the inpaintingprocess, which is accomplished with the method presented in [5].

Using the same methodology as in the proof of Theorem 1.1 we obtain the following existencetheorem,

Theorem 1.4. Let f ∈ L2(Ω). The stationary equation

∆p+ λ(f − u) = 0, p ∈ ∂TV (u) (1.7)

admits a solution u ∈ BV (Ω).We shall also give a characterization of elements in the subdifferential ∂TV (u) for TV (u)

defined as in (1.6), i.e., TV (u) = |Du| (Ω) + χ1(u), where

χ1(u) =

0 if |u| ≤ 1 a.e. in Ω+∞ otherwise.


Namely, we shall prove the following theorem.Theorem 1.5. Let p ∈ ∂χ1(u). An element p ∈ ∂TV (u) with |u(x)| ≤ 1 in Ω, fulfills the

following set of equations

p = −∇ ·(∇u|∇u|

)a.e. on supp (|u| < 1)

p = −∇ ·(∇u|∇u|

)+ p, p ≤ 0 a.e. on supp (|u| = −1)

p = −∇ ·(∇u|∇u|

)+ p, p ≥ 0 a.e. on supp (|u| = 1) .

For (1.5) we additionally give error estimates for the inpainting error and stability informationin terms of the Bregman distance. Let ftrue be the original image and u a stationary solution of(1.5). In our considerations we use the symmetric Bregman distance defined as

DsymmTV (u, ftrue) = 〈u− ftrue, p− ξ〉 , p ∈ ∂TV (u), ξ ∈ ∂TV (ftrue).

We prove the following resultTheorem 1.6. Let ftrue fulfill the so called source condition, i.e.,

There exists ξ ∈ ∂TV (ftrue) such that ξ = ∆−1q for a source element q ∈ H−1(Ω),

and u ∈ BV (Ω) be a stationary solution of (1.5) given by u = us + ud, where us is a smoothfunction and ud is a piecewise constant function. Then the inpainting error reads

DsymmTV (u, ftrue) +

λ0

2‖u− ftrue‖2−1 ≤

1λ0

‖ξ‖21 + Cλ0 |D|(r−2)/r errinpaint,

with constant C > 0 and

errinpaint := K1 +K2

(|D|C (M(us), β) + 2

∣∣R(ud)∣∣) ,

where K1,K2, and C are appropriate positive constants, M(us) is the smoothness bound for us,β is determined from the shape of D, and the error region R(ud) is defined from the level lines ofud.

Finally we present numerical results for the proposed binary- and grayvalue inpainting ap-proaches and briefly explain the numerical implementation using convexity splitting methods.

Organization of the paper. In Section 2 a fixed point approach is proposed to proveTheorem 1.1, i.e., the existence of a stationary solution for the modified Cahn-Hilliard equation(1.1) with Dirichlet boundary conditions. In Section 3 we discuss the new TV −H−1 inpaintingapproach. We prove Theorem 1.2 in Subsection 3.1. This Γ−limit is generalized to an inpaintingapproach for grayvalue images, called TV −H−1 inpainting (cf. (1.5)). Similarly to the existenceproof in Section 2 we prove in Subsection 3.2 the existence of a stationary solution of this newinpainting approach for grayvalue images, i.e., Theorem 1.4. In Section 3.3 we additionally givea characterization of elements in the subdifferential of the corresponding regularizing functional(1.6). In addition we present error estimates for both the error in inpainting the image by meansof (1.5) and for the stability of solutions of (1.5) in terms of the Bregman distance, i.e., theproof of Theorem 1.6, in Subsection 3.4. Section 4 is dedicated to the numerical solution ofCahn-Hilliard- and TV − H−1−inpainting and the presentation of numerical examples. Finallyin Appendix A the space H−1

∂ is defined and its elements are characterized in order to analyze(1.1) for Neumann boundary conditions. In Appendix B basic facts about functions of boundedvariation are presented.

Notation. Before we begin with the discussion of our results let us introduce some no-tations. By ‖.‖ we always denote the norm in L2(Ω) with corresponding inner product 〈., .〉and by ‖.‖−1 :=

∥∥∇∆−1.∥∥ the norm in H−1(Ω) = (H1

0 (Ω))∗ with corresponding inner product〈., .〉−1 :=

⟨∇∆−1.,∇∆−1.

⟩where ∆−1 denotes the inverse of the negative Laplacian −∆ on H1

0 ,i.e., ∆−1 : H−1(Ω) → H1

0 (Ω).


2. Cahn-Hilliard inpainting - proof of Theorem 1.1. In this chapter we prove theexistence of a weak solution of the stationary equation (1.3). Let Ω ⊂ R2 be a bounded Lipschitzdomain and f ∈ L2(Ω) given. In order to be able to impose boundary conditions in the equation,we assume f to be constant in a small neighborhood of ∂Ω. This assumption is for technicalpurposes only and does not influence the inpainting process as long as the inpainting domain Ddoes not touch the boundary of the image domain Ω. Instead of Neumann boundary data as inthe original Cahn-Hilliard inpainting approach (cf. [8]) we use Dirichlet boundary conditions forour analysis, i.e., we consider

ut = ∆(−ε∆u+ 1

εF′(u)

)+ λ(f − u) in Ω

u = f,−ε∆u+ 1εF

′(u) = 0 on ∂Ω. (2.1)

This change from a Neumann- to a Dirichlet problem makes it easier to deal with the boundaryconditions in our proofs but does not have a significant impact on the inpainting process as long aswe assume that D ⊂ Ω. In Appendix A we nevertheless propose a setting to extend the presentedanalysis for (1.1) to the originally proposed model with Neumann boundary data. In our new set-ting we define a weak solution of equation (1.3) as a function u ∈ H =

u ∈ H1(Ω), u|∂Ω = f |∂Ω

that fulfills

〈ε∇u,∇φ〉+⟨

1εF ′(u), φ

⟩− 〈λ(f − u), φ〉−1 = 0, ∀φ ∈ H1

0 (Ω). (2.2)

Remark 2.1. With u ∈ H1(Ω) and the compact embedding H1(Ω) →→ Lq(Ω) for every1 ≤ q <∞ and Ω ⊂ R2 the weak formulation is well defined.

To see that (2.2) defines a weak formulation for (1.3) with Dirichlet boundary conditions weintegrate by parts in (2.2) and get∫

Ω(−ε∆u+ 1

εF′(u)−∆−1 (λ(f − u))) φ dx

−∫

∂Ω∆−1 (λ(f − u))∇∆−1φ · ν dH1 = 0, ∀φ ∈ H1

0 (Ω),

where H1 denotes the one dimensional Hausdorff measure. This yieldsε∆u− 1

εF′(u) + ∆−1 (λ(f − u)) = 0 in Ω

∆−1 (λ(f − u)) = 0 on ∂Ω. (2.3)

Assuming sufficient regularity on u we can use the definition of ∆−1 to see that u solves−ε∆∆u+ 1

ε ∆F ′(u) + λ(f − u) = 0 in Ω∆−1 (λ(f − u)) = −ε∆u+ 1

εF′(u) = 0 on ∂Ω,

Since additionally u|∂Ω = f |∂Ω, the function u solves (1.3) with Dirichlet boundary conditions.For the proof of existence of a solution to (2.2) we follow the subsequent strategy. We consider

the fixed point operator A : L2(Ω) → L2(Ω) where A(v) = u fulfills for a given v ∈ L2(Ω) theequation

1τ ∆−1(u− v) = ε∆u− 1

εF′(u) + ∆−1 [λ(f − u) + (λ0 − λ)(v − u)] in Ω,

u = f, ∆−1(

1τ (u− v) + λ(f − u) + (λ0 − λ)(v − u)

)= 0 on ∂Ω, (2.4)

where τ > 0 is a parameter. The boundary conditions of A are given by the second equation in(2.4). Note that actually the solution u will be in H1(Ω) and hence the boundary condition iswell-defined in the trace sense, the operator A into L2(Ω) is then obtained with further embedding.We define a weak solution of (2.4) as before by a function u ∈ H =

u ∈ H1(Ω), u|∂Ω = f |∂Ω

that fulfills ⟨

1τ (u− v), φ

⟩−1

+ 〈ε∇u,∇φ〉+⟨

1εF

′(u), φ⟩

−〈λ(f − u) + (λ0 − λ)(v − u), φ〉−1 = 0 ∀φ ∈ H10 (Ω).

(2.5)


A fixed point of the operator A, provided it exists, then solves the stationary equation withDirichlet boundary conditions as in (2.3).

Note that in (2.4) the characteristic function λ in the fitting term λ(f −u)+(λ0−λ)(v−u) =λ0(v−u)+λ(f−v) only appears in combination with given functions f, v and is not combined withthe solution u of the equation. For equation (2.4), i.e., (2.5), we can therefore state a variationalformulation. This is, for a given v ∈ L2(Ω) equation (2.4) is the Euler-Lagrange equation of theminimization problem

u∗ = argminu∈H1(Ω),u|∂Ω=f |∂ΩJε(u, v) (2.6)

with

Jε(u, v) =∫

Ω

(ε

2|∇u|2 +

1εF (u)

)dx+

12τ

‖u− v‖2−1 +λ0

2

∥∥∥∥u− λ

λ0f −

(1− λ

λ0

)v

∥∥∥∥2

−1

. (2.7)

We are going to use the variational formulation (2.7) to prove that (2.4) admits a weak solutionin H1(Ω). This solution is unique under additional conditions.

Proposition 2.2. Equation (2.4) admits a weak solution in H1(Ω) in the sense of (2.5).For τ ≤ Cε3, where C is a positive constant depending on |Ω|, |D|, and F only, the weak solutionof (2.4) is unique.

Further we prove that the operator A admits a fixed point under certain conditions.Proposition 2.3. Set A : L2(Ω) → L2(Ω), A(v) = u, where u ∈ H1(Ω) is the unique weak

solution of (2.4). Then A admits a fixed point u ∈ H1(Ω) if τ ≤ Cε3 and λ0 ≥ C 1ε3 for a positive

constant C depending on |Ω|, |D|, and F only.Hence the existence of a stationary solution of (1.1) follows under the condition λ0 ≥ C/ε3.We begin with considering the fixed point equation (2.4), i.e., the minimization problem (2.6).

In the following we prove the existence of a unique weak solution of (2.4) by showing the existenceof a unique minimizer for (2.7).

Proof. (Proof of Proposition 2.2) We want to show that Jε(u, v) has a minimizer in H =u ∈ H1(Ω), u|∂Ω = f |∂Ω

. For this we consider a minimizing sequence un ∈ H of Jε(u, v). To

see that un is uniformly bounded in H1(Ω) we show that Jε(u, v) is coercive in H1(Ω). WithF (u) ≥ C1u

2 − C2 for two positive constants C1, C2 > 0 and the triangular inequality in theH−1(Ω) space, we obtain

Jε(u, v) ≥ ε

2‖∇u‖22 +

C1

ε‖u‖22 −

C2

ε+

12τ

(12‖u‖2−1 − ‖v‖2−1

)+λ0

2

(12‖u‖2−1 −

∥∥∥∥ λλ0f +

(1− λ

λ0

)v

∥∥∥∥2

−1

)

≥ ε

2‖∇u‖22 +

C1

ε‖u‖22 +

(λ0

4+

14τ

)‖u‖2−1 − C3(v, f, λ0, ε,Ω, D).

Therefore a minimizing sequence un is bounded in H1(Ω) and it follows that un u∗ inH1(Ω). To finish the proof of existence for (2.4) we have to show that Jε(u, v) is weakly lowersemicontinuous in H1(Ω). For this we divide the sequence Jε(un, v) of (2.7) in two parts. Wedenote the first term

an =∫

Ω

(ε

2|∇un|2 +

1εF (un)

)dx︸︷︷︸

CH(un)

and the second term

bn =12τ

‖un − v‖2−1︸︷︷︸D(un,v)

+λ0

2

∥∥∥∥un − λ

λ0f −

(1− λ

λ0

)v

∥∥∥∥2

−1︸︷︷︸FIT (un,v)

.


Since H1 →→ L2 it follows un → u∗ in L2(Ω). Further we know that if bn converges strongly,then

lim inf (an + bn) = lim inf an + lim bn. (2.8)

We begin with the consideration of the last term in (2.7). We denote f := λλ0f + (1 − λ

λ0)v.

We want to show ∥∥∥un − f∥∥∥2

−1−→

∥∥∥u∗ − f∥∥∥2

−1⇐⇒

〈∆−1(un − f), un − f〉 −→ 〈∆−1(u∗ − f), u∗ − f〉.

For this we consider the absolute difference of the two terms,

|〈∆−1(un − f), un − f〉 − 〈∆−1(u∗ − f), u∗ − f〉|= |〈∆−1(un − u∗), un − f〉 − 〈∆−1(u∗ − f), un − u∗〉|≤ |〈un − u∗,∆−1(un − f)〉|+ |〈∆−1(u∗ − f), u∗ − un〉|

≤ ‖un − u∗‖︸︷︷︸→0

·∥∥∥∆−1(un − f)

∥∥∥+ ‖un − u∗‖︸︷︷︸→0

·∥∥∥∆−1(u∗ − f)

∥∥∥Since the operator ∆−1 : H−1(Ω) → H1

0 (Ω) is a linear and continuous operator it follows that∥∥∆−1F∥∥ ≤ ∥∥∆−1

∥∥ · ‖F‖ for all F ∈ H−1(Ω).

Thus

|〈∆−1(un − f), un − f〉 − 〈∆−1(u∗ − f), u∗ − f〉|

≤ ‖un − u∗‖︸︷︷︸→0

∥∥∆−1∥∥︸︷︷︸

const

∥∥∥un − f∥∥∥︸︷︷︸

bounded

+ ‖un − u∗‖︸︷︷︸→0

∥∥∆−1∥∥︸︷︷︸

const

∥∥∥u∗ − f∥∥∥︸︷︷︸

const

−→ 0 as n→∞,

and we conclude that FIT (un, v) converges strongly to FIT (u∗, v). With the same argument itfollows that D(un, v) converges strongly and in sum that the sequence bn converges strongly inL2(Ω). Further CH(.) is weakly lower semicontinuous, which follows from the lower semicontinuityof the Dirichlet integral and from the continuity of F by applying Fatou’s Lemma. Hence we obtain

Jε(u∗, v) ≤ lim inf Jε(un, v).

Therefore Jε has a minimizer in H1, i.e.,

∃u∗ with u∗ = argminu∈H1(Ω)Jε(u, v).

We next assert that u∗ fulfills the boundary condition u∗|∂Ω = f |∂Ω.To see this, note that for anadmissible function w ∈ H, un − w ∈ H1

0 (Ω). Now H10 (Ω) is a closed, linear subspace of H1(Ω),

and so, by Mazur’s theorem (cf. [23] § D.4 for example), is weakly closed. Hence u∗ −w ∈ H10 (Ω)

and consequently the trace of u∗ on ∂Ω is equal to f .For simplicity let in the following u = u∗. To see that the minimizer u is a weak solution of

(2.4) we compute the corresponding Euler-Lagrange equation to the minimization problem. Forthis sake we choose any test function φ ∈ H1

0 (Ω) and compute the first variation of Jε, i.e.,(d

dδJε(u+ δφ, v)

)δ=0

,


which has to be zero for a minimizer u. Thus we have

ε 〈∇u,∇φ〉+1ε〈F ′(u), φ〉+

⟨1τ

(u− v) + λ0

[u− λ

λ0f −

(1− λ

λ0

)v

], φ

⟩−1

= 0.

Integrating by parts in both terms we get⟨−ε∆u+

1εF ′(u)−∆−1

(1τ

(u− v) + λ0

[u− λ

λ0f −

(1− λ

λ0

)v

]), φ

⟩+∫

∂Ω

∇u · νφ ds+∫

∂Ω

∆−1

(1τ

(u− v) + λ0

[u− λ

λ0f −

(1− λ

λ0

)v

])∇∆−1φ · ν ds = 0.

Since φ is an element in H10 (Ω) the first boundary integral vanishes. Further a minimizer u fulfills

the boundary condition u = f on the boundary ∂Ω. Hence, we obtain that u fulfills the weakformulation (2.5) of (2.4).

For the uniqueness of the minimizer, we need to prove that Jε is strictly convex. To do so,we prove that for any u1, u2 ∈ H1(Ω),

Jε(u1, v) + Jε(u2, v)− 2Jε

(u1 + u2

2, v

)> 0, (2.9)

based on an assumption that F (.) satisfies F (u1) + F (u2) − 2F (u1+u22 ) > −C(u1 − u2)2, for a

constant C > 0. For example, when F (u) = 18 (u2 − 1)2, C = 1

8 . Denote u = u1 − u2, we have

Jε(u1, v) + Jε(u2, v)− 2Jε

(u1 + u2

2, v

)>ε

4‖u‖2H1 +

(14τ

+λ0

4

)‖u‖2−1 −

C

ε‖u‖22

By using the inequality

‖u‖22 ≤ ‖u‖H1 ‖u‖−1 , (2.10)

and the Cauchy-Schwarz inequality, for (2.9) to be fulfilled, we need

2

√ε

4

(14τ

+λ0

4

)≥ C

ε.

i.e.,

ε3(λ0 +

1τ

)≥ C2.

Therefore Jε(u, v) is strictly convex in u and our minimization problem has a unique minimizer ifτ is chosen smaller than Cε3 for a constant C depending on |Ω|, |D|, and F only. Because of theconvexity of Jε in ∇u and u, every weak solution of the Euler-Lagrange equation (2.4) is in facta minimizer of Jε. This proves the uniqueness of a weak solution of (2.4) provided τ << Cε3.

Next we want to prove Proposition 2.3, i.e., the existence of a fixed point of (2.4) and withthis the existence of a stationary solution of (1.1). To do so we are going to apply Schauder’s fixedpoint theorem.

Proof. (Proof of Proposition 2.3) We consider a solution A(v) = u of (2.4) with v ∈ L2(Ω)given. In the following we will prove the existence of a fixed point by using Schauder’s fixed pointtheorem. We start with proving that

‖A(v)‖2 = ‖u‖2 ≤ β ‖v‖2 + α, (2.11)

for a constant β < 1. Having this we have shown that A is a map from the closed ball K =B(0,M) =

u ∈ L2(Ω) : ‖u‖ ≤M

into itself for an appropriate constant M > 0. We conclude


the proof with showing the compactness of K and the continuity of the fixed point operator A.From Schauder’s theorem the existence of a fixed point follows.

Let us, for the time being, additionally assume that ∇u and ∆u are bounded in L2(Ω). Hencewe can multiply (2.4) with −∆u and integrate over Ω to obtain

−∫

Ω

∆u∆−1

[1τ

(u− v)− λ(f − u)− (λ0 − λ)(v − u)]dx = −ε 〈∆u,∆u〉+

1ε

∫Ω

F ′(u)∆u dx

After integration by parts we find with the short-hand notation

w :=1τ

(u− v)− λ(f − u)− (λ0 − λ)(v − u)

that∫Ω

uw dx−∫

∂Ω

[∇u · ν(∆−1w +

1εF ′(u))− u∇(∆−1w) · ν

]dH1 = −ε ‖∆u‖2 − 1

ε

∫Ω

F ′′(u) |∇u|2 dx.

Now we insert the boundary conditions ∆−1w = 0, u = f =: f1 and F ′(u) = F ′(f) = f2 on ∂Ωwith constants f1 and f2 on the left-hand side, i.e.∫

Ω

uw dx−∫

∂Ω

[f2ε∇u · ν − f1∇(∆−1w) · ν

]dH1 = −ε ‖∆u‖2 − 1

ε

∫Ω

F ′′(u) |∇u|2 dx.

An application of Gauss’ Theorem to the boundary integral implies∫∂Ω

[f2ε∇u · ν − f1∇(∆−1w) · ν

]dH1 =

f2ε

∫Ω

∆u dx+ f1

∫Ω

w dx,

and we get∫Ω

uw dx = −ε ‖∆u‖2 − 1ε

∫Ω

F ′′(u) |∇u|2 dx+f2ε

∫Ω

∆u dx+ f1

∫Ω

w dx.

By further applying Young’s inequality to the last two terms we get∫Ω

uw dx ≤(f2δ

2ε− ε

)‖∆u‖2 − 1

ε

∫Ω

F ′′(u) |∇u|2 dx+f1δ

2‖w‖2 + C(f1, f2, |Ω|, ε, δ).

Using the identity λ(f − u) + (λ0 − λ)(v − u) = λ(f − v) + λ0(v − u) in the definition of w yields∫Ω

u · 1τ

(u− v) ≤(f2δ

2ε− ε

)‖∆u‖2 − 1

ε

∫Ω

F ′′(u) |∇u|2 dx+f1δ

2‖w‖2

+λ0

(∫Ω\D

u(f − u) dx+∫

D

u(v − u) dx

)+ C(f1, f2, |Ω|, ε, δ).

By applying the standard inequality (a+ b)2 ≤ 2(a2 + b2) to the L2 norm of w = ( 1τ +λ0)u− ( 1

τ +λ0 − λ)v − λf and by using that (1− λ/λ0) ≤ 1 in the resulting L2 norm of v we get∫

Ω

u · 1τ

(u− v) ≤(f2δ

2ε− ε

)‖∆u‖2 − 1

ε

∫Ω

F ′′(u) |∇u|2 dx+ f1δ

(1τ

+ λ0

)2

‖u‖2

+2f1δ(

1τ

+ λ0

)2

‖v‖2 + λ0

(∫Ω\D

u(f − u) dx+∫

D

u(v − u) dx

)+C(f, f1, f2, |Ω|, ε, δ, λ0).


With F ′′(u) ≥ C1u2−C2 for some constants C1, C2 > 0 and for all u ∈ R, and by further applying

the Cauchy-Schwarz inequality to the last two integrals we obtain∫Ω

u · 1τ

(u− v) ≤(f2δ

2ε− ε

)‖∆u‖2 − C1

ε‖u |∇u|‖2 +

C2

ε‖∇u‖2 + f1δ

(1τ

+ λ0

)2

‖u‖2

+2f1δ(

1τ

+ λ0

)2

‖v‖2 + λ0

[−(

1− δ22

)∫Ω\D

u2 dx+(δ12− 1)∫

D

u2 dx

+1

2δ1

∫D

v2 dx

]+ C(f, f1, f2, |Ω|, |D|, ε, λ0, δ, δ2).

Setting δ2 = 1 and δ1 = 2 we see that∫Ω

u · 1τ

(u− v) ≤(f2δ

2ε− ε

)‖∆u‖2 − C1

ε‖u |∇u|‖2 +

C2

ε‖∇u‖2 + f1δ

(1τ

+ λ0

)2

‖u‖2

+2f1δ(

1τ

+ λ0

)2

‖v‖2 + λ0

[−1

2

∫Ω\D

u2 dx+14

∫D

v2 dx

]+C(f, f1, f2, |Ω|, |D|, ε, δ, λ0).

We follow the argumentation of the proof of existence for (1.1) in [8] by observing the followingproperty: A standard interpolation inequality for ∇u reads

‖∇u‖2 ≤ δ3 ‖∆u‖2 +C3

δ3‖u‖2 . (2.12)

The domain of integration in the second integral of the equation above can be taken to be smallerthan Ω by taking a larger constant C3. Further we use the L1 version of Poincare’s inequalityapplied to the function u2. We recall this inequality in the following theorem.

Theorem 2.4. (Poincare’s inequality in L1). Assume that Ω is precompact open subset ofn-dimensional Euclidean space Rn having Lipschitz boundary (i.e., Ω is an open, bounded Lipschitzdomain). Then there exists a constant C, depending only on Ω, such that, for every function u inthe Sobolev space W 1,1(Ω),

‖u− uΩ‖L1(Ω) ≤ C‖∇u‖L1(Ω),

where uΩ = 1|Ω|∫Ωu(y) dy is the average value of u over Ω, with |Ω| denoting the Lebesgue measure

of the domain Ω.Then, assuming that u 6= 0 in Ω \D, we choose the constant C4 (which depends on the size of Dcompared to Ω) large enough such that∫

Ω

u2 dx− C4

∫Ω\D

u2 dx ≤∫

Ω

∣∣u2 − u2Ω

∣∣ dx ≤ C4

∫Ω

∣∣∇u2∣∣ dx.

or in other words

‖u‖2 ≤ C4

∥∥∇u2∥∥

L1(Ω)+ C4

∫Ω\D

u2 dx. (2.13)

By Holders inequality we also have that∥∥∇u2∥∥

L1(Ω)≤ α

2‖u |∇u|‖2 +

C5

2α. (2.14)

Putting the last three inequalities (2.12)-(2.14) together we obtain

‖∇u‖2 ≤ δ3 ‖∆u‖2 +C3C4α

2δ3‖u |∇u|‖2 +

C3C4

δ3

∫Ω\D

u2 dx+C3C4C5

2αδ3.


We now use the last inequality to bound the gradient term in our estimates from above to get∫Ωu · 1

τ (u− v) ≤ ( f2δ+2C2δ32ε − ε) ‖∆u‖2 + (C2C3C4α

2δ3ε − C1ε ) ‖u |∇u|‖2

+(f1δ(

1τ + λ0

)2 + C2C3C4δ3ε − C4λ0

2 ) ‖u‖2 +(

λ04 + 2f2δ

(1τ + λ0

)2) ‖v‖2+C(f, f1, f2, |Ω|, |D|, ε, δ, λ0).

(2.15)With δ3 < 2ε2−f2δ

2C2and α, δ small enough the first two terms can be estimated from above by zero.

Applying the Cauchy-Schwarz inequality on the left-hand side and rearranging the terms on bothsides of the inequality we conclude(

12τ

+C4λ0

2− f1δ

(1τ− λ0

)2

− C2C3C4

δ3ε

)‖u‖2 ≤

((λ0

4+

12τ

)+ 2f2δ

(1τ

+ λ0

)2)‖v‖2

+C(f, f1, f2, |Ω|, |D|, ε, δ, λ0).

Choosing δ small enough, C4 large enough, and λ0 ≥ CC41ε3 the solution u and v fulfill

‖u‖2 ≤ β ‖v‖2 + C, (2.16)

with β < 1 and a constant C independent of v. Hence u is bounded in L2(Ω).To see that our regularity assumptions on u from the beginning of the proof are automatically

fulfilled we consider (2.15) with appropriate constants δ3, δ, and α as specified in the paragraphbelow (2.15). But now we only estimate the second term on the right side by zero and keep thefirst term. By applying the Cauchy-Schwarz inequality and rearranging the terms as before weobtain (

12τ + C4λ0

2 − f1δ(

1τ − λ0

)2 − C2C3C4δ3ε

)‖u‖2 + (ε− f2δ+2C2δ3

2ε ) ‖∆u‖2

≤((

λ04 + 1

2τ

)+ 2f2δ

(1τ + λ0

)2) ‖v‖2 + C(f, f1, f2, |Ω|, |D|, ε, δ, λ0),

with the coefficient ε− f2δ+2C2δ32ε > 0 due to our choice of δ3. Therefore not only the L2− norm of

u is uniformly bounded but also the L2− norm of ∆u. By the standard interpolation inequality(2.12) the boundedness of u in H1(Ω) follows. From the last result we additionally get that theoperator A is a compact map since A : L2(Ω) → H1(Ω) →→ L2(Ω). Therefore K is a compactand convex subset of L2(Ω).It remains to show that the operator A is continuous. Indeed if vk → v in L2(Ω) then A(vk) = uk isbounded in H1(Ω) for all k = 0, 1, 2, . . .. Thus, we can consider a weakly convergent subsequenceukj

u in H1(Ω). Because H1(Ω) →→ Lq(Ω), 1 ≤ q < ∞ the sequence ukjconverges also

strongly to u in Lq(Ω). Hence, a weak solution A(vk) = uk of (2.4) weakly converges to a weaksolution u of

1τ

(−∆−1)(u− v) = ε∆u− 1εF ′(u)−∆−1 [λ(f − u) + (λ0 − λ)(v − u)] ,

where u is the weak limit of A(vk) as k → ∞. Because the solution of (2.4) is unique providedτ ≤ Cε3 (cf. Proposition 2.2), u = A(v), and therefore A is continuous. Applying Schauder’sTheorem we have shown that the fixed point operator A admits a fixed point u in L2(Ω) whichfulfills

〈ε∇u,∇φ〉+⟨

1εF ′(u), φ

⟩−〈λ(f − u), φ〉−1+

∫∂Ω

∆−1 (λ(f − u))∇∆−1φ·ν dH1 = 0, ∀φ ∈ H10 (Ω).

Because the solution of (2.4) is an element of H =u ∈ H1(Ω), u|∂Ω = f |∂Ω

also the fixed point

u ∈ H.Following the arguments from the beginning of this section we conclude with the existence of

a stationary solution for (1.1).


By modifying the setting and the above proof in an appropriate way one can prove the existenceof a stationary solution for (1.1) also under Neumann boundary conditions, i.e.,

∇u · ν = ∇∆u · ν = 0, on ∂Ω.

A corresponding reformulation of the problem is given in Appendix A.

3. Total Variation - H−1 inpainting. In this section we discuss our newly proposed in-painting scheme (1.5), i.e., the inpainted image u of f ∈ L2(Ω) evolves via

ut = ∆p+ λ(f − u), p ∈ ∂TV (u),

with

TV (u) =


Before starting this section we suggest readers who are unfamiliar with the space BV (Ω) to firstread Appendix B and maybe recall the definition of the subdifferential of a function in DefinitionB.9.

3.1. Γ-Convergence of the Cahn-Hilliard energy - proof of Theorem 1.2. In thefollowing we want to motivate our new inpainting approach (1.5) by considering the Γ−limit forε→ 0 of an appropriate time-discrete Cahn-Hilliard inpainting approach, i.e., the Γ− limit of thefunctionals from our fixed point approach in (2.7). More precisely we want to prove Theorem 1.2stated in the Introduction of this paper. Before starting our discussion lets recall the definitionof Γ-convergence and its impact within the study of optimization problems. For more details onΓ-convergence we refer to [29].

Definition 3.1. Let X = (X, d) be a metric space and (Fh), h ∈ N be family of functionsFh : X → [0,+∞]. We say that (Fh) Γ-converges to a function F : X → [0,+∞] on X if ∀x ∈ Xwe have

(i) for every sequence xh with d(xh, x) → 0 we have

F (x) ≤ lim infh

Fh(xh);

(ii) there exists a sequence xh such that d(xh, x) → 0 and

F (x) = limhFh(xh)

(or, equivalently, F (x) ≥ lim suph Fh(xh)).We write F (x) = Γ − limh Fh(x), x ∈ X, is the Γ-limit of (Fh) in X. The formulation of theΓ-limit for ε→ 0 is analogous by defining a sequence εh with εh → 0 as h→∞.

The important property of Γ-convergent sequences of functions Fh is that its minima convergeto minima of the Γ-limit F . In fact we have the following theorem

Theorem 3.2. Let (Fh) be like in Definition 3.1 and additionally equicoercive, that is thereexists a compact set K ⊂ X (independent of h) such that

infx∈X

Fh(x) = infx∈K

Fh(x).

If Fh Γ-converges on X to a function F we have

minx∈X

F (x) = limh

infx∈X

Fh(x) .

After recalling these facts about Γ-convergence we continue this section with the proof ofTheorem 1.2.


Proof. Modica and Mortola have shown in [31] and [32] that the sequence of Cahn-Hilliardfunctionals

CH(u) =∫

Ω

(ε

2|∇u|2 +

1εF (u)

)dx

Γ-converges in the topology L1(Ω) to

TV (u) =

C0 |Du| (Ω) if u = χE for some Borel measurable subset E ⊂ Ω+∞ otherwise

as ε → 0, where C0 = 2∫ 1

0

√F (s) ds. (The space BV (Ω) and the total variation |Du| (Ω) are

defined in Appendix B.)Now, for a given function v ∈ L2(Ω) the functional Jε from our fixed point approach (2.4), i.e.,

Jε(u, v) =∫

Ω

(ε

2|∇u|2 +

1εF (u)

)dx︸︷︷︸

:=CH(u)

+12τ

‖u− v‖2−1︸︷︷︸:=D(u,v)

+λ0

2

∥∥∥∥u− λ

λ0f − (1− λ

λ0)v∥∥∥∥2

−1︸︷︷︸:=FIT (u,v)

,

is the sum of the regularizing term CH(u), the damping term D(u, v) and the fitting termFIT (u, v). We recall the following fact,

Theorem 3.3. [Dal Maso, [29], Prop. 6.21.] Let G : X → R be a continuous function and(Fh) Γ− converges to F in X, then (Fh +G) Γ− converges to F +G in X.Since the H−1-norm is continuous in H−1(Ω) and hence in particular in L1(Ω), the two terms inJε that are independent from ε, i.e., D(u, v) and FIT (u, v), are continuous in L1(Ω). Togetherwith the Γ-convergence result of Modica and Mortola for the Cahn-Hilliard energy, we have proventhat the modified Cahn-Hilliard functional Jε can be seen as a regularized approximation in thesense of Γ-convergence of the TV-functional

J(u, v) = TV (u) +D(u, v) + FIT (u, v),

for functions u ∈ BV (Ω) with u(x) = χE for a Borel measurable subset E ⊂ Ω. In fact we havegone from a smooth transition layer between 0 and 1 in the Cahn-Hilliard inpainting approach(depending on the size of ε) to a sharp interface limit in which the image function now jumps from0 to 1.

This property motivates the extension of J(u, v) to grayvalue functions such that |u| ≤ 1on Ω and hence leads us from the Cahn-Hilliard inpainting approach for binary images to ageneralization for grayvalue images, namely our so called TV −H−1 inpainting equation (1.5).

3.2. Existence of a stationary solution - proof of Theorem 1.4. Our strategy forproving the existence of a stationary solution for TV − H−1 inpainting (1.5) is similar to ourexistence proof for a stationary solution of the modified Cahn-Hilliard equation (1.1) in Section2. Similarly as in our analysis for (1.1) in Section 2 we consider equation (1.5) with Dirichletboundary conditions, namely

ut = ∆p+ λ(f − u) in Ωu = f on ∂Ω,

for p ∈ ∂TV (u).Now let f ∈ L2(Ω), |f | ≤ 1 be the given grayvalue image. For v ∈ Lr(Ω), 1 < r < 2, we

consider the minimization problem

u∗ = arg minu∈BV (Ω)

J(u, v),

with functionals

J(u, v) := TV (u) +12τ||u− v||2−1︸︷︷︸D(u,v)

+λ0

2||u− λ

λ0f − (1− λ

λ0)v||2−1︸︷︷︸

FIT (u,v)

, (3.1)


with TV (u) defined as in (1.6), i.e.,

TV (u) =


Note that Lr(Ω) can be continuously embedded in H−1(Ω). Hence the functionals in (3.1) arewell defined.

First we will show that for a given v ∈ Lr(Ω) the functional J(., v) attains a unique minimizeru∗ ∈ BV (Ω) with |u∗(x)| ≤ 1 a.e. in Ω.

Proposition 3.4. Let f ∈ L2(Ω) be given with |f(x)| ≤ 1 a.e. in Ω and v ∈ Lr(Ω). Thenthe functional J(., v) has a unique minimizer u∗ ∈ BV (Ω) with |u∗(x)| ≤ 1 a.e. in Ω.

Proof. Let (un)n∈N be a minimizing sequence for J(u, v), i.e.,

J(un, v) → infu∈BV (Ω)

J(u, v).

Then un ∈ BV (Ω) and |un(x)| ≤ 1 in Ω (because otherwise TV (un) would not be finite). Therefore

|Dun| (Ω) ≤M, for an M ≥ 0 and for all n ≥ 1,

and, because of the uniform boundedness of |u(x)| for every point x ∈ Ω,

‖un‖Lp(Ω) ≤ M, for an M ≥ 0, ∀n ≥ 1, and 1 ≤ p ≤ ∞.

Thus un is uniformly bounded in Lp(Ω) and in particular in L1(Ω). Together with the boundednessof |Dun| (Ω), the sequence un is also bounded in BV (Ω) and there exists a subsequence, stilldenoted un, and a u ∈ BV (Ω) such that un u weakly in Lp(Ω), 1 ≤ p ≤ ∞ and weakly∗ inBV (Ω). Because L2(Ω) ⊂ L2(R2) ⊂ H−1(Ω) (by zero extensions of functions on Ω to R2) un ualso weakly in H−1(Ω). Because |Du| (Ω) is lower semicontinuous in BV (Ω) and by the lowersemicontinuity of the H−1 norm we get

J(u, v)= TV (u) +D(u, v) + FIT (u, v)≤ lim infn→∞(TV (un) +D(un, v) + FIT (un, v))= lim infn→∞J(un, v).

So u is a minimizer of J(u, v) over BV (Ω).

To prove the uniqueness of the minimizer we (similarly as in the proof of Theorem 2.2) showthat J is strictly convex. Namely we prove that for all u1, u2 ∈ BV (Ω), u1 6= u2

J(u1, v) + J(u2, v)− 2J(u1 + u2

2, v

)> 0.

We have

J(u1, v) + J(u2, v)− 2J(u1 + u2

2, v

)=(

12τ

+λ0

2

)(‖u1‖2−1 + ‖u2‖2−1 − 2

∥∥∥∥u1 + u2

2

∥∥∥∥2

−1

)

+TV (u1) + TV (u2)− 2TV(u1 + u2

2

)≥(

14τ

+λ0

4

)‖u1 − u2‖2−1 > 0.

This finishes the proof.Next we shall prove the existence of stationary solution for (1.5). For this sake we consider

the corresponding Euler-Lagrange equation to (3.1), i.e.,

∆−1

(u− v

τ

)+ p−∆−1 (λ(f − u) + (λ0 − λ)(v − u)) = 0,


with the weak formulation⟨1τ (u− v), φ

⟩−1

+ 〈p, φ〉 − 〈λ(f − u) + (λ0 − λ)(v − u), φ〉−1 ∀φ ∈ H10 (Ω).

A fixed point of the above equation, i.e., a solution u = v, is then a stationary solution for (1.5).Thus, to prove the existence of a stationary solution of (1.5), i.e., to prove Theorem 1.4, we asbefore are going to use a fixed point argument. Let A : Lr(Ω) → Lr(Ω), 1 < r < 2, be the operatorwhich maps a given v ∈ Lr(Ω) to A(v) = u under the condition that A(v) = u is the minimizerof the functional J(., v) defined in (3.1). The choice of the fixed point operator A over Lr(Ω) wasmade in order to obtain the necessary compactness properties for the application of Schauder’stheorem.

Since here the treatment of the boundary conditions is similar as in Section 2 we will leavethis part of the analysis in the upcoming proof to the reader and just carry out the proof withoutexplicitly taking care of the boundary.

Proof. Let A : Lr(Ω) → Lr(Ω), 1 < r < 2, be the operator that maps a given v ∈ Lr(Ω) toA(v) = u, where u is the unique minimizer of the functional J(., v) defined in (3.1). Existenceand uniqueness follow from Theorem 3.4. Since u minimizes J(., v) we have u ∈ L∞(Ω) henceu ∈ Lr(Ω). Additionally we have J(u, v) ≤ J(0, v), i.e.,

12τ ||u− v||2−1 + λ0

2 ||u−λλ0f − (1− λ

λ0)v||2−1 + TV (u) ≤ 1

2τ ||v||2−1 + λ0

2 ||λλ0f + (1− λ

λ0)v||2−1

≤ |Ω|2τ + λ0(|Ω|+ |D|).

(3.2)Here the last inequality was obtained since Lr(Ω) → H−1(Ω) and hence ||v||−1 ≤ C and ||λv||−1 ≤C for a C > 0. (In fact, since H1(Ω) → Lr′(Ω) for all 1 ≤ r′ < ∞ from duality it followsthat Lr(Ω) → H−1(Ω) for, 1 < r < ∞.) By the last estimate we obtain u ∈ BV (Ω). SinceBV (Ω) →→ Lr(Ω) compactly for 1 ≤ r ≤ 2 and Ω ⊂ R2 (cf. Theorem B.7), the operator A mapsLr(Ω) → BV (Ω) →→ Lr(Ω), i.e., A : Lr(Ω) → K, where K is a compact subset of Lr(Ω). Thus,for v ∈ B(0, 1) (where B(0, 1) denotes the ball in L∞(Ω) with center 0 and radius 1), the operatorA : B(0, 1) → B(0, 1) ∩K = K, where K is a compact and convex subset of Lr(Ω).

Next we have to show that A is continuous in Lr(Ω). Let (vk)k≥0 be a sequence whichconverges to v in Lr(Ω). Then uk = A(vk) solves

∆pk =uk − vk

τ− (λ(f − uk) + (λ0 − λ)(vk − uk)) ,

where pk ∈ ∂TV (uk). Thus uk is uniformly bounded in BV (Ω) ∩ L∞(Ω) (and hence in Lr(Ω))and, since the right-hand side of the above equation is uniformly bounded in Lr(Ω), also ∆pk isbounded in Lr(Ω). Thus there exists a subsequence pkl

such that ∆pkl ∆p in Lr(Ω) and a

subsequence uklthat converges weakly ∗ to a u in BV (Ω)∩L∞(Ω). Since BV (Ω) →→ Lr(Ω) we

have ukl→ u strongly in Lr(Ω). Therefore the limit u solves

∆p =u− v

τ− (λ(f − u) + (λ0 − λ)(v − u)) . (3.3)

If we additionally apply Poincare’s inequality to ∆pk we conclude

‖∇pk − (∇pk)Ω‖Lr(Ω)≤ C ‖∇ · (∇pk − (∇pk)Ω)‖

Lr(Ω),

where (∇pk)Ω = 1|Ω|∫Ω∇pk dx. In addition, since pk ∈ ∂TV (uk), it follows that (pk)Ω = 0 and

‖pk‖BV ∗(Ω) ≤ 1. Thus (∇pk)Ω <∞ and pk is uniformly bounded in W 1,r(Ω). Thus there exists asubsequence pkl

such that pkl p in W 1,r(Ω). In addition Lr′(Ω) → BV ∗(Ω) for 2 < r′ <∞ (this

follows again from Theorem B.7 by a duality argument) and W 1,r(Ω) →→ Lq(Ω) for 1 ≤ q < 2r2−r

(Rellich-Kondrachov Compactness Theorem, cf. [2], Theorem 8.7). By choosing 2 < q < 2r2−r we

have in sum W 1,r(Ω) →→ BV ∗(Ω). Thus pkl→ p strongly in BV ∗(Ω). Hence the element p in

(3.3) is an element in ∂TV (u).Because the minimizer of (3.1) is unique, u = A(v), and therefore A is continuous in Lr(Ω).

From Schauder’s fixed point theorem the existence of a stationary solution follows.


3.3. Characterization of Solutions - proof of Theorem 1.5. Finally we want to computeelements p ∈ ∂TV (u). Like in [13] the model for the regularizing functional is the sum of astandard regularizer plus the indicator function of the L∞ constraint. Especially we have TV (u) =|Du| (Ω) + χ1(u), where |Du| (Ω) is the total variation of Du and

χ1(u) =

0 if |u| ≤ 1 a.e. in Ω+∞ otherwise.

(3.4)

We want to compute the subgradients of TV by pretending ∂TV (u) = ∂ |Du| (Ω) + ∂χ1(u). Thismeans we can separately compute the subgradients of χ1. To guarantee that the splitting above is

allowed we have to consider a regularized functional of the total variation, like∫Ω

√|∇u|2 + δ dx.

This is sufficient because both |D.| (Ω) and χ1 are convex and |D.| (Ω) is continuous (compare [21]Proposition 5.6., pp. 26).

The subgradient ∂ |Du| (Ω) is already well described, as, for instance, in [4] or [37]. We willjust shortly recall its characterization. Thereby we do not insist on the details of the rigorousderivation of these conditions, and we limit ourself to mention the main facts.

It is well known [37, Proposition 4.1] that p ∈ ∂|Du|(Ω) impliesp = −∇ · ( ∇u

|∇u| ) in Ω∇u|∇u| · ν = 0 on ∂Ω.

The previous conditions do not fully characterize p ∈ ∂|Du|(Ω), additional conditions would berequired [4, 37], but the latter are, unfortunately, hardly numerically implementable. Since weanyway consider a regularized version of |Du| (Ω) the subdifferential becomes a gradient whichreads p = −∇ · ( ∇u√

|∇u|2+δ) in Ω

∇u√|∇u|2+δ

· ν = 0 on ∂Ω.

The subgradient of χ1 is computed like in the following Lemma.Lemma 3.5. Let χ1 : Lr(Ω) → R ∪ ∞ be defined by (3.4), and let 1 ≤ r ≤ ∞. Then

p ∈ Lr∗(Ω), for r∗ = rr−1 , is a subgradient p ∈ ∂χ1(u) for u ∈ Lr(Ω) with χ1(u) = 0, if and only

if

p = 0 a.e. on supp(|u| < 1)p ≤ 0 a.e. on supp(u = −1)p ≥ 0 a.e. on supp(u = 1).

Proof. Let p ∈ ∂χ1(u). Then we can choose v = u + εw for w being any bounded functionsupported in |u| < 1− α for arbitrary 0 < α < 1. If ε is sufficiently small we have |v| ≤ 1.Hence

0 ≥ 〈v − u, p〉 = ε

∫|u|<1−α

wp dx.

Since we can choose ε both positive and negative, we obtain∫|u|<1−α

wp dx = 0.

Because 0 < α < 1 and w are arbitrary we conclude p = 0 on the support of |u| < 1. If wechoose v = u+ w with w is an arbitrary bounded function with

0 ≤ w ≤ 1 on supp(−1 ≤ u ≤ 0)w = 0 on supp(0 < u ≤ 1).


Then v is still between −1 and 1 and

0 ≥ 〈v − u, p〉 =∫u=−1

wp dx+∫u=1

wp dx

=∫u=−1

wp dx.

Because w is arbitrary and positive on u = −1 it follows that p ≤ 0 a.e. on u = −1. If wechoose now v = u+ w with w is an arbitrary bounded function with

w = 0 on supp(−1 ≤ u ≤ 0)−1 ≤ w ≤ 0 on supp(0 < u ≤ 1).

Then v is still between −1 and 1 and

0 ≥ 〈v − u, p〉 =∫u=−1

wp dx+∫u=1

wp dx

=∫u=1

wp dx.

Analogue to before, since w is arbitrary and negative on u = 1 it follows that p ≥ 0 a.e. onu = 1.On the other hand assume that

p = 0 a.e. on supp(|u| < 1)p ≤ 0 a.e. on supp(u = −1)p ≥ 0 a.e. on supp(u = 1).

We need to verify the subgradient property

〈v − u, p〉 ≤ χ1(v)− χ1(u) = χ1(v) for all v ∈ Lr(Ω)

only for χ1(v) = 0, since it is trivial for χ1(v) = ∞. So let v ∈ Lr(Ω) be a function between −1and 1 almost everywhere on Ω. Then with p as above we obtain

〈v − u, p〉 =∫u=−1

p(v − u) dx+∫u=1

p(v − u) dx

=∫u=−1

p(v + 1) dx+∫u=1

p(v − 1) dx.

Since −1 ≤ v ≤ 1 the first and the second term are always ≤ 0 since p ≤ 0 for u = −1 and p ≥ 0for u = 1 respectively. Therefore 〈v − u, p〉 ≤ 0 and we are done.

3.4. Error estimation and stability analysis with the Bregman distance - proofof Theorem 1.6. In the following analysis we want to present estimates for both the error weactually make in inpainting an image with our TV −H−1 approach (1.5) (see (3.12)) and for thestability of solutions for this problem (see (3.13)) in terms of the Bregman distance. This section ismotivated by the error analysis for variational models in image restoration with Bregman iterationsin [13], and the error estimates for inpainting models developed in [15]. In [13] the authors consideramong other things the general optimality condition

p+ λ0A∗(Au− fdam) = 0, (3.5)

where p ∈ ∂R(u) for a regularizing term R, A is a bounded linear operator and A∗ its adjoint.Now the error that is to be estimated depends on the form of smoothing of the image containedin (3.5). Considering this equation one realizes that the smoothing consists of two steps. The first


smoothing is created by the operator A which depends on the image restoration task at hand,and is actually a dual one that smooths the subgradient p. The second smoothing step is theone, which is directly implied by the regularizing term, i.e., its subgradient p, and depends on therelationship between the primal variable u and the dual variable p. A condition that representsthis dual smoothing property of functions, i.e., subgradients, is the so-called source condition. Letftrue be the original image then the source condition for ftrue reads:

There exists ξ ∈ ∂R(ftrue) such that ξ = A∗q for a source element q ∈ D(A∗), (3.6)

where D(A∗) is the domain of the operator A∗. It can be shown (cf. [12]) that this is equivalentto require from ftrue to be a minimizer of

R(u) +λ0

2‖Au− fdam‖2 ,

for arbitrary fdam ∈ D(A∗) and λ0 ∈ R. Now, the source condition has a direct consequence forthe Bregman distance, which gives rise to its use for the subsequent error analysis. To be moreprecise, the Bregman distance is defined as

DpR(v, u) = R(v)−R(u)− 〈v − u, p〉 , p ∈ ∂R(u).

Then, if ftrue fulfills the source condition with a particular subgradient ξ we obtain

DξR(u, ftrue) = R(u)−R(ftrue)− 〈u− ftrue, ξ〉 = R(u)−R(ftrue)− 〈q, Au−Aftrue〉 ,

and thus the Bregman distance can be both related to the error in the regularization functional(R(u)−R(ftrue)) and the output error (Au−Aftrue). For the sake of symmetry properties in thesequel we shall consider the symmetric Bregman distance, which is defined as

DsymmR (u1, u2) = Dp1

R (u2, u1) +Dp2R (u1, u2) = 〈u1 − u2, p1 − p2〉 , pi ∈ ∂R(ui).

Additionally to this error analysis we shall get a control on the inpainting error |u− ftrue| insidethe inpainting domain D by means of estimates from [15]. Therein the authors analyzed theinpainting process by understanding how the regularizer continues level lines into the missingdomain. The inpainting error was then determined by means of the definition of an error region,smoothness bounds on the level lines, and quantities taking into the account the shape of theinpainting domain. In the following we are going to implement both strategies, i.e., [13] and [15],in order to proof Theorem 1.6.

Proof. Let fdam ∈ L2(Ω) be the given damaged image with inpainting domain D ⊂ Ω andftrue the original image. We consider the stationary equation to (1.5), i.e.,

−∆p+ λ(u− fdam) = 0, p ∈ ∂TV (u), (3.7)

where we define TV (u) as a functional over L2(Ω) as

TV (u) =

|Du| (Ω) if u ∈ BV (Ω), ‖u‖L∞ ≤ 1+∞ otherwise.

In the subsequent we want to characterize the error we make by solving (3.7) for u, i.e., how largedo we expect the distance between the restored image u and the original image ftrue to be.

Now, let ∆−1 be the inverse operator to −∆ with zero Dirichlet boundary conditions as before.In our case the operator A in (3.5) is the embedding operator from H1

0 (Ω) into H−1(Ω) and standsin front of the whole term A(u− fdam), cf. (3.7). The adjoint operator is A∗ = ∆−1 which mapsH−1(Ω) into H1

0 (Ω). We assume that the given image fdam coincides with ftrue outside of theinpainting domain, i.e.,

fdam = ftrue in Ω \Dfdam = 0 in D. (3.8)


Further we assume that ftrue satisfies the source condition (3.6), i.e.,

There exists ξ ∈ ∂TV (ftrue) such that ξ = A∗q = ∆−1q for a source element q ∈ H−1(Ω). (3.9)

For the following analysis we first rewrite (3.7). For u, a solution of (3.7), we get

p+ λ0∆−1(u− ftrue) = ∆−1 [(λ0 − λ)(u− ftrue)] , p ∈ ∂TV (u).

Here we replaced fdam by ftrue using assumption (3.8). By adding a ξ ∈ ∂TV (ftrue) from (3.9)to the above equation we obtain

p− ξ + λ0∆−1(u− ftrue) = −ξ + λ0∆−1

[(1− λ

λ0

)(u− ftrue)

]Taking the duality product with u− ftrue (which is just the inner product in L2(Ω) in our case)we get

DsymmTV (u, ftrue) + λ0 ‖u− ftrue‖2−1 =

⟨∇ξ,∇∆−1(u− ftrue)

⟩+ λ0

⟨(1− λ

λ0

)(u− ftrue), u− ftrue

⟩−1

,

where

DsymmTV (u, ftrue) = 〈u− ftrue, p− ξ〉 , p ∈ ∂TV (u), ξ ∈ ∂TV (ftrue).

An application of Young’s inequality yields


λ0

2‖u− ftrue‖2−1 ≤

1λ0

‖ξ‖21 + λ0

∥∥∥∥(1− λ

λ0

)(u− ftrue)

∥∥∥∥2

−1

(3.10)

For the last term we obtain∥∥∥(1− λλ0

)v∥∥∥−1

= supφ,‖φ‖−1=1

⟨φ,(1− λ

λ0

)v⟩−1

= supφ,‖φ‖−1=1−⟨∆−1φ,

(1− λ

λ0

)v⟩

= supφ,‖φ‖−1=1−⟨(

1− λλ0

)∆−1φ, v

⟩≤Holder ‖v‖2 · supφ,‖φ‖−1=1

∥∥∥(1− λλ0

)∆−1φ

∥∥∥2.

With ∆−1 : H−1 → H1 → Lr, 2 < r <∞ we get

∫Ω

((1− λ

λ0

)∆−1φ

)2

dx =∫

D

(∆−1φ

)2dx ≤Holder |D|

1q′ ·(∫

Ω

(∆−1φ

)2q) 1

q

=choose q= r2|D| r−2

r ·∥∥∆−1φ

∥∥2

p≤H1→Lr C|D| r−2

r ‖φ‖2−1 = C|D| r−2r ,

i.e., ∥∥∥∥(1− λ

λ0

)v

∥∥∥∥2

−1

≤ C|D|r−2

r ‖v‖2 . (3.11)

Applying (3.11) to (3.10) we see that


λ0

2‖u− ftrue‖2−1 ≤

1λ0

‖ξ‖21 + Cλ0 |D|(r−2)/r ‖u− ftrue‖2

To estimate the last term we use some error estimates for TV− inpainting computed in [15]. Firstwe have

‖u− ftrue‖2 =∫

Ω\D(u− ftrue)2 dx+

∫D

(u− ftrue)2 dx.


Since u − ftrue is uniformly bounded in Ω (this follows from the L∞ bound in the definition ofTV (u)) we estimate the first term by a positive constant K1 and the second term by the L1 normover D. We obtain

‖u− ftrue‖2 ≤ K1 +K2

∫D

|u− ftrue| dx.

Now let u ∈ BV (Ω) be given by u = us + ud, where us is a smooth function and ud is a piecewiseconstant function. Following the error analysis in [15] (Theorem 8.) for functions u ∈ BV (Ω) wehave

‖u− ftrue‖2 ≤ K1 +K2err(D)≤ K1 +K2

(|D|C (M(us), β) + 2

∣∣R(ud)∣∣) ,

where M(us) is the smoothness bound for us, β is determined from the shape of D, and the errorregion R(ud) is defined from the level lines of ud. Note that in general the error region fromhigher-order inpainting models including the TV seminorm is smaller than that from TV − L2

inpainting (cf. Section 3.2. in [15]).Finally we end up with


λ0

2‖u− ftrue‖2−1 ≤

1λ0

‖ξ‖21 + Cλ0 |D|(r−2)/r errinpaint, (3.12)

with

errinpaint := K1 +K2

(|D|C (M(us), β) + 2

∣∣R(ud)∣∣) .

The first term in (3.12) depends on the regularizer TV , and the second term on the size of theinpainting domain D.

Remark 3.6. From inequality (3.12) we derive an optimal scaling for λ0, i.e., a scaling whichminimizes the inpainting error. It reads

λ20 |D|

r−2r ∼ 1

λ0 ∼ |D|−r−22r .

In two space dimensions r can be chosen arbitrarily big, which gives λ0 ∼ 1/√|D| as the optimal

order for λ0.Stability estimates for (3.7) can also be derived with an analogous technique. For ui being

the solution of (3.7) with fdam = fi (again assuming that fi = ftrue in Ω \D), the estimate

DsymmJ (u1, u2) +

λ0

2‖u1 − u2‖2−1 ≤

λ0

2

∫D

(f1 − f2)2 dx (3.13)

holds.

4. Numerics. In the following numerical results for the two inpainting approaches (1.1) and(1.5) are presented. For both approaches we used convexity splitting algorithms, proposed by Eyrein [24], for the discretization in time. For more details to the application of convexity splittingalgorithms in higher order inpainting compare [10].

For the space discretization we used the cosine transform to compute the finite differences forthe derivatives in a fast way and to preserve the Neumann boundary conditions in our inpaintingapproaches (also cf. [10] for a detailed description).

4.1. Convexity splitting scheme for Cahn-Hilliard inpainting. For the discretizationin time we use a convexity splitting scheme applied by Bertozzi et al. [8] to Cahn-Hilliard inpaint-ing. The original Cahn-Hilliard equation is a gradient flow in H−1 for the energy

E1[u] =∫

Ω

ε

2|∇u|2 +

1εF (u) dx,


Fig. 4.1. Destroyed binary image and the solution of Cahn-Hilliard inpainting with switching ε value: u(1200)with ε = 0.1, u(2400) with ε = 0.01



while the fitting term in (1.1) can be derived from a gradient flow in L2 for the energy

E2[u] =12

∫Ω

λ(f − u)2 dx.

We apply convexity splitting for both E1 and E2 separately. Namely we split E1 as E1 = E11−E12

with

E11 =∫

Ω

ε

2|∇u|2 +

C1

2|u|2 dx,

and

E12 =∫

Ω

−1εF (u) +

C1

2|u|2 dx.

A possible splitting for E2 is E2 = E21 − E22 with

E21 =12

∫Ω

C2

2|u|2 dx,


and

E22 =12

∫Ω

−λ(f − u)2 +C2

2|u|2 dx.

For the splittings discussed above the resulting time-stepping scheme is

uk+1 − uk

τ= −∇H−1(Ek+1

11 − Ek12)−∇L2(Ek+1

12 − Ek22),

where ∇H−1 and ∇L2 represent gradient descent with respect to the H−1 inner product and theL2 inner product respectively. This translates to a numerical scheme of the form

uk+1 − uk

τ+ ε∆∆uk+1 − C1∆uk+1 + C2uk+1 =

1ε∆F ′(uk)− C1∆uk + λ(f − uk) + C2uk.(4.1)

To make sure that E11, E12 and E21, E22 are convex the constants C1 >1ε , C2 > λ0.

For the discretization in space we used finite differences and spectral methods, i.e., the discretecosine transform, to symplify the inversion of the Laplacian ∆ for the computation of uk+1.

In [10] the authors prove that the above timestepping scheme is unconditionally stable in thesense that the numerical solution uk is uniformly bounded on a finite time interval. Moreoverthe discrete solution converges to the exact solution of (1.1) as τ → 0. These properties make(4.1) a stable and reliable discrete approximation of the continuous equation (1.1). Concerningthe performance of the scheme in terms of computational speed its already remarked in [7] and[8] that (4.1) is certainly faster than numerical solutions for competing curvature based models.Nevertheless the convexity conditions on the constants C1 and C2 damp the convergence of thescheme and hence the solution of the scheme to steady state may require a lot of iterations (cf. alsothe number of iterations needed to compute the examples in Figures 4.1-4.3). An investigationof fast numerical solvers for (1.1) is a matter of future research. A very recent approach in thisdirection was made in [11], where the authors propose a multigrid approach for inpainting withCDD.

In Figures 4.1-4.3 Cahn-Hilliard inpainting was applied to three different binary images. Inall of the examples we follow the procedure of [7], i.e., the inpainted image is computed in a twostep process. In the first step Cahn-Hilliard inpainting is solved with a rather large value of ε,e.g., ε = 0.1, until the numerical scheme is close to steady state. In this step the level lines arecontinued into the missing domain. In a second step the result of the first step is put as an initialcondition into (4.1) for a small ε, e.g., ε = 0.01, in order to sharpen the contours of the imagecontents. The reason for this two step procedure is twofold. First of all in [8] the authors givenumerical evidence that the steady state of (2.4) is not unique, i.e., it is dependent on the initialcondition for the equation. As a consequence, computing the inpainted image by the application ofCahn-Hilliard inpainting with a small ε only, might not prolongate the level lines into the missingdomain as desired. See also [8] for a bifurcation diagram based on the numerical computationsof the authors. The second reason for solving Cahn-Hilliard inpainting in two steps is that it iscomputationally less expensive. Solving (4.1) for, e.g., ε = 0.1 is faster than solving it for ε = 0.01.Again, this is because of the damping in (4.1) introduced by the constant C1.

4.2. Convexity splitting scheme for TV −H−1 inpainting. We consider equation (1.5)where p ∈ ∂TV (u) is replaced by the formal expression ∇ · ( ∇u

|∇u| ), namely

ut = −∆(∇ · ( ∇u|∇u|

)) + λ(f − u). (4.2)

Similar to the convexity splitting for the Cahn-Hilliard inpainting we propose the following splittingfor the TV-H−1 inpainting equation. The regularizing term in (4.2) can be modeled by a gradientflow in H−1 of the energy

E1 =∫

Ω

|∇u| dx.


We split E1 in E11 − E12 with

E11 =∫

Ω

C1

2|∇u|2 dx

E12 =∫

Ω

−|∇u|+ C1

2|∇u|2 dx.

The fitting term is a gradient flow in L2 of the energy

E2 =12

∫Ω

λ(f − u)2dx

and is splitted into E2 = E21 − E22 with

E21 =∫

Ω

C2

2|u|2 dx

E22 =12

∫Ω

−λ(f − u)2 + C2|u|2 dx.

Analogous to above the resulting time-stepping scheme is

uk+1 − uk

τ+ C1∆∆uk+1 + C2uk+1 = C1∆∆uk −∆(∇ · ( ∇uk

|∇uk|)) + C2uk + λ(f − uk). (4.3)

In order to make the scheme unconditionally stable, the constants C1 and C2 have to be chosenso that E11, E12, E21, E22 are all convex. The choice of C1 depends on the regularization of the

total variation we are using. Using the square regularization |∇u| is replaced by√|∇u|2 + δ2 the

condition turns out to be C1 >1δ and C2 > λ0.

The discrete scheme (4.3) was analyzed in [10]. As for Cahn-Hilliard inpainting, thereinthe authors prove that the scheme is unconditionally stable. For this case the convergence ofthe discrete solution to the exact solution of (1.5) (where the subgradient p was replaced by itsrelaxed version ∇ · (∇u/

√|∇u|2 + δ2)) only holds under additional assumptions on the regularity

of the exact solution. Results for the one dimensional case developed in [9] suggest that theseregularity assumptions also hold in two dimensions with a sufficiently regular initial conditionfor the equation. However a rigorous proof of this fact is currently missing and is a challengeof additional research. As in the case of Cahn-Hilliard inpainting the convexity splitting scheme(4.3) for TV-H−1 inpainting converges rather slow due to the damping induced by C1 and C2.Nevertheless its solution is numerically stable and approximates the exact solution accurately(which was rigorously proven for smooth solutions of (1.5) in [10]).

In Figures 4.4-4.8 examples for the application of TV-H−1 inpainting to grayvalue images areshown. In Figure 4.5 a comparison of the TV-H−1 inpainting result with the result obtained bythe second order TV-L2 inpainting model for a crop of the image in Figure 4.4 is presented. Thesuperiority of the fourth-order TV-H−1 inpainting model to the second order model with respectto the desired continuation of edges into the missing domain is clearly visible. Other exampleswhich support this claim are presented in Figure 4.6 and 4.7 where the line is connected by theTV-H−1 inpainting model but clearly splitted by the second-order TV-L2 model. It would beinteresting to strengthen this numerical observation with a rigorous result as it was done in [8] forCahn-Hilliard inpainting, cf. (1.2). The authors consider this as another important contributionof future research.

Acknowledgments. This work was partially supported by KAUST (King Abdullah Uni-versity of Science and Technology), by the WWTF (Wiener Wissenschafts-, Forschungs- undTechnologiefonds) project nr.CI06 003, by the FFG project Erarbeitung neuer Algorithmen zumImage Inpainting project nr. 813610, and the PhD program Wissenschaftskolleg taking place atthe University of Vienna.


Fig. 4.4. TV-H−1 inpainting: u(1000) with λ0 = 103

Fig. 4.5. (l.) u(1000) with TV-H−1 inpainting, (r.) u(5000) with TV-L2 inpainting

Fig. 4.6. TV-H−1 inpainting compared to TV-L2 inpainting: u(5000) with λ0 = 10

The authors further would like to thank Andrea Bertozzi for her suggestions concerning thefixed point approach for the stationary equation, Massimo Fornasier for several discussions on thetopic and Peter Markowich for remarks on the manuscript. We would also like to thank the editorand the referees for useful comments.


Fig. 4.7. TV-H−1 inpainting compared to TV-L2 inpainting: u(5000) with λ0 = 10

Fig. 4.8. TV −H−1 inpainting: u(1000) with λ = 103

Appendix A. Neumann boundary conditions and the space H−1∂ (Ω).

In this section we want to pose the Cahn-Hilliard inpainting problem with Neumann boundaryconditions in a way such that the analysis from Section 2 can be carried out in a similar way.Namely we consider

ut = ∆(−ε∆u+ 1εF

′(u)) + λ(f − u) in Ω,∂u∂ν = ∂∆u

∂ν = 0 on ∂Ω,

For the existence of a stationary solution of this equation we consider again a fixed point approachsimilar to (2.4) in the case of Dirichlet boundary conditions, i.e.,

u−vτ = ∆(−ε∆u+ 1

εF′(u)) + λ(f − u) + (λ0 − λ)(v − u) in Ω,

∂u∂ν =

∂(ε∆u− 1ε F ′(u))

∂ν = 0 on ∂Ω.(A.1)

To reformulate the above equation in terms of the operator ∆−1 with Neumann boundary condi-tions we first have to introduce the space H−1

∂ (Ω) in which the operator ∆−1 is now the inverseof −∆ with Neumann boundary conditions.


Thus we define the non-standard Hilbert space

H−1∂ (Ω) =

F ∈ H1(Ω)∗ | 〈F, 1〉(H1)∗,H1 = 0

.

Since Ω is bounded we know 1 ∈ H1(Ω), hence H−1∂ (Ω) is well defined. Before we define a norm

and an inner product on H−1∂ (Ω) we have to define more spaces. Let

H1φ(Ω) =

ψ ∈ H1(Ω) :

∫Ω

ψ dx = 0,

with norm ‖u‖H1φ

:= ‖∇u‖L2 and inner product 〈u, v〉H1φ

:= 〈∇u,∇v〉L2 . This is a Hilbert space

and the norms ‖.‖H1 and ‖.‖H1φ

are equivalent on H1φ(Ω). Let (H1

φ(Ω))∗ denote the dual of H1φ(Ω).

We will use (H1φ(Ω))∗ to induce an inner product on H−1

∂ (Ω). Given F ∈ (H1φ(Ω))∗ with associate

u ∈ H1φ(Ω) (from the Riesz representation theorem) we have by definition

〈F,ψ〉(H1φ)∗,H1

φ= 〈u, ψ〉H1

φ= 〈∇u,∇ψ〉L2 ∀ψ ∈ H1

φ(Ω).

Lets now define a norm and an inner product on H−1∂ (Ω).

Definition A.1.

H−1∂ (Ω) :=

F ∈ H1(Ω)∗ | 〈F, 1〉(H1)∗,H1 = 0

‖F‖H−1

∂:=∥∥F |H1

φ

∥∥(H1

φ)∗

〈F1, F2〉H−1∂

:= 〈∇u1,∇u2〉L2 ,

where F1, F2 ∈ H−1∂ (Ω) and where u1, u2 ∈ H1

φ(Ω) are the associates of F1|H1φ, F2|H1

φ ∈(H1

φ(Ω))∗.At this point it is not entirely obvious that for a given F ∈ H−1

∂ (Ω) we have F |H1φ ∈ (H1

φ(Ω))∗.That this is the case though is explained in the following theorem.

Theorem A.2.1. H−1

∂ (Ω) is closed in (H1(Ω))∗.2. The norms ‖.‖H−1

∂and ‖.‖(H1)∗ are equivalent on H−1

∂ (Ω).Theorem A.2 can be easily checked just by the application of the definitions and the fact that

the norms ‖.‖H1 and ‖.‖H1φ

are equivalent on H1φ(Ω). From point 1. of the theorem we have that

H−1∂ (Ω) is a Hilbert space w.r.t. the (H1(Ω))∗ norm and point 2. tells us that the norms ‖.‖H−1

∂

and ‖.‖(H1)∗ are equivalent on H−1∂ (Ω). Therefore the norm in Definition A.1 is well defined and

H−1∂ (Ω) is a Hilbert space w.r.t. ‖.‖H−1

∂.

In the following we want to characterize elements F ∈ H−1∂ (Ω). By the above definition we

have for each F ∈ H−1∂ (Ω), there exists a unique element u ∈ H1

φ(Ω) such that

〈F,ψ〉(H1)∗,H1 =∫

Ω

∇u · ∇ψ dx, ∀ψ ∈ H1φ(Ω). (A.2)

Since 〈F, 1〉(H1)∗,H1 = 0, we see that 〈F,ψ +K〉(H1)∗,H1 = 〈F,ψ〉(H1)∗,H1 for all constants K ∈ Rand therefore (A.2) extends to all ψ ∈ H1(Ω). We define

∆−1F := u (A.3)

the unique solution to (A.2).Now suppose F ∈ L2(Ω) and assume u ∈ H2(Ω). Set 〈F,ψ〉 :=

∫ΩFψ dx. Because L2(Ω) ⊂

H−1∂ (Ω) an element F is also an element in H−1

∂ (Ω). Thus there exists a unique element u ∈ H1φ(Ω)

such that ∫Ω

(−∆u− F )ψ dx+∫

∂Ω

∇u · νψ ds = 0, ∀ψ ∈ H1φ(Ω).


Therefore u ∈ H1φ(Ω) is the unique weak solution of the following problem:

−∆u− F = 0 in Ω∇u · ν = 0 on ∂Ω. (A.4)

Remark A.3. With the above characterization of elements F ∈ H−1∂ (Ω) and the notation

(A.3) for its associates the inner product and the norm can be written as

〈F1, F2〉H−1∂

:=∫

Ω

∇∆−1F1 · ∇∆−1F2 dx, ∀F1, F2 ∈ H−1∂ (Ω),

and norm

‖F‖H−1∂

:=

√∫Ω

(∇∆−1F )2 dx.

Throughout the rest of this appendix we will write the short forms 〈., .〉−1 and ‖.‖−1 for the innerproduct and the norm in H−1

∂ (Ω) respectively.Now its important to notice that in order to rewrite (A.1) in terms of ∆−1 we require the

”right hand side” of the equation, i.e., u−vτ + λ(u− f) + (λ0 − λ)(u− v) to be an element of our

new space H−1∂ (Ω) (cf. Definition A.1). In other words the ”right hand side” has to have zero

mean over Ω. Because we cannot guarantee this property for solutions of the fixed point equation(A.1) we are going to modify the right hand side by subtracting its mean. Let

FΩ = 1τ F

1Ω + λ0F

2Ω

F 1Ω = 1

|Ω|∫Ω(u− v) dx

F 2Ω = 1

|Ω|∫Ω

λλ0

(u− f) +(1− λ

λ0

)(u− v) dx,

and consider instead of (A.1) the equationε∆u− 1

εF′(u) = ∆−1

(u−v

τ − λ(f − u)− (λ0 − λ)(v − u)− FΩ

)in Ω,

∂u∂ν = 0 on ∂Ω,

where the second Neumann boundary condition∂(ε∆u− 1

ε F ′(u))∂ν = 0 on ∂Ω is included in the

definition of ∆−1. The functional of the corresponding variational formulation then reads

Jε(u, v) =∫Ω

(ε2 |∇u|

2 + 1εF (u)

)dx+ 1

2τ

∥∥(u− v)− F 1Ω

∥∥2

−1

+λ02

∥∥∥(u− λλ0f −

(1− λ

λ0

)v)− F 2

Ω

∥∥∥2

−1.

With these definitions the proof for the existence of a stationary solution for the modified Cahn-Hilliard equation with Neumann boundary conditions can be carried out similarly to the proof inSection 2. Note that every solution of (1.1) fulfills d

dt

∫Ωu dx =

∫Ωλ(f − u) dx. This means that

for a stationary solution u the integral C∫Ωλ(f − u) dx = 0 for every constant C ∈ R (,i.e., the

”right hand side” has zero mean and therefore F 1Ω = F 2

Ω = 0).

Appendix B. Functions of bounded variation.The following results can be found in [3]. Let Ω ⊂ R2 be an open and bounded Lipschitz

domain. As in [3] the space of functions of bounded variation BV (Ω) in two space dimensions isdefined as follows:

Definition B.1. (BV (Ω)) Let u ∈ L1(Ω). We say that u is a function of boundedvariation in Ω if the distributional derivative of u is representable by a finite Radon measure inΩ, i.e., if ∫

Ω

u∂φ

∂xidx = −

∫Ω

φdDiu ∀φ ∈ C∞c (Ω), i = 1, 2,


for some R2− valued measure Du = (D1u,D2u) in Ω. The vector space of all functions of boundedvariation in Ω is denoted by BV (Ω). Further, the space BV (Ω) can be characterized by the totalvariation of Du. For this we first define the so called variation V (u,Ω) of a function u ∈ L1

loc(Ω).Definition B.2. (Variation) Let u ∈ L1

loc(Ω). The variation V (u,Ω) of u in Ω is definedby

V (u,Ω) := sup∫

Ω

u divφ dx : φ ∈(C1

c (Ω))2, ‖φ‖∞ ≤ 1

.

A simple integration by parts proves that

V (u,Ω) =∫

Ω

|∇u| dx,

if u ∈ C1(Ω). By a standard density argument this is also true for functions u ∈W 1,1(Ω). Beforewe proceed with the characterization of BV (Ω) let us recall the definition of the total variation ofa measure:

Definition B.3. (Total variation of a measure) Let (X, E) be a measure space. If µ is ameasure, we define its total variation |µ| as follows:

|µ| (E) := sup

∞∑h=0

|µ(Eh)| : Eh ∈ E pairwise disjoint , E =∞⋃

h=0

Eh

, ∀E ⊂ E .

With Definition B.2 the space BV (Ω) can be characterized as followsTheorem B.4. Let u ∈ L1(Ω). Then, u belongs to BV (Ω) if and only if V (u,Ω) < ∞.

In addition, V (u,Ω) coincides with |Du| (Ω), the total variation of Du, for any u ∈ BV (Ω) andu 7→ |Du| (Ω) is lower semicontinuous in BV (Ω) with respect to the L1

loc(Ω) topology.Note that BV (Ω) is a Banach space with respect to the norm

‖u‖BV (Ω) = ‖u‖L1(Ω) + |Du| (Ω).

Now we introduce so called weak∗ convergence in BV (Ω) which is useful for its compactnessproperties. Note that this convergence is much weaker than the norm convergence.

Definition B.5. (Weak∗ convergence) Let u, uh ∈ BV (Ω). We say that (uh) weakly∗

converges in BV (Ω) to u (in signs uh∗ u) if (uh) converges to u in L1(Ω) and (Duh) weakly∗

converges to Du in all (Ω), i.e.,

limh→∞

∫Ω

φ dDuh =∫

Ω

φ dDu ∀φ ∈ C0(Ω).

A simple criterion for weak∗ convergence is the following:Theorem B.6. Let (uh) ⊂ BV (Ω). Then (uh) weakly∗ converges to u in BV (Ω) if and only

if (uh) is bounded in BV (Ω) and converges to u in L1(Ω).Further we have the following compactness theorem:Theorem B.7. (Compactness for BV (Ω))• Let Ω be a bounded domain with compact Lipschitz boundary. Every sequence (uh) ⊂BVloc(Ω) satisfying

sup

∫A

|uh| dx+ |Duh| (A) : h ∈ N<∞ ∀A ⊂⊂ Ω open,

admits a subsequence (uhk) converging in L1

loc(Ω) to u ∈ BVloc(Ω). If the sequence isfurther bounded in BV (Ω) then u ∈ BV (Ω) and a subsequence converges weakly∗ to u.

• Let Ω be a bounded domain in Rd with Lipschitz boundary. Then, every uniformly boundedsequence (uk)k≥0 in BV (Ω) is relatively compact in Lr(Ω) for 1 ≤ r < d

d−1 , d ≥ 1.Moreover, there exists a subsequence ukj and u in BV (Ω) such that ukj u weakly∗ inBV (Ω). In particular for d = 2 this compact embedding holds for 1 ≤ r < 2.


Let u ∈ L1(Ω). We introduce the mean value uΩ of u as

uΩ :=1|Ω|

∫Ω

u(x) dx.

A generalization of the Poincare inequality gives the so called Poincare-Wirtinger inequality forfunctions in BV (Ω).

Theorem B.8. (Poincare-Wirtinger inequality) If Ω ⊂ R2 is a bounded, open and connecteddomain with compact Lipschitz boundary, we have

‖u− uΩ‖Lp(Ω) ≤ C |Du| (Ω) ∀u ∈ BV (Ω), 1 ≤ p ≤ 2

for some constant C depending only on Ω.Finally we introduce the notion of the subdifferential of a function.Definition B.9. Let X be a locally convex space, X∗ its dual, 〈., .〉 the bilinear pairing over

X ×X∗ and F a mapping of X into R. The subdifferential of F at u ∈ X is defined as

∂F (u) = p ∈ X∗| 〈v − u, p〉 ≤ F (v)− F (u),∀v ∈ X .

Every normed vector space is a locally convex space and therefore the theory of subdifferentialsapplies to our framework (with X = BV (Ω)).

REFERENCES

[1] R. Acar, and C.R. Vogel, Analysis of bounded variation penalty method for ill-posed problems, Inverse Prob-lems, Vol. 10, No. 6, pp.1217-1229, 1994.

[2] H.W. Alt, Linear Funktionalanalysis, Springer Verlag, 1992.[3] L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems,

Mathematical Monographs, Oxford University Press, 2000.[4] G. Aubert, and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations

and the Calculus of Variations, Springer Verlag, Applied Mathematical Sciences, Vol 147, November2001.

[5] M. Bertalmio, G. Sapiro, V. Caselles, and C. Ballester, Image inpainting, Computer Graphics, SIGGRAPH2000, July, 2000.

[6] M. Bertalmio, L. Vese, G. Sapiro, and S. Osher, Simultaneous structure and texture image inpainting, IEEETrans. Image Process., Vol. 12, Nr. 8, pp. 882-889, 2003.

[7] A. Bertozzi, S. Esedoglu, and A. Gillette, Inpainting of Binary Images Using the Cahn-Hilliard Equation,IEEE Trans. Image Proc. 16(1) pp. 285-291, 2007.

[8] A. Bertozzi, S. Esedoglu, and A. Gillette, Analysis of a two-scale Cahn-Hilliard model for image inpainting,Multiscale Modeling and Simulation, vol. 6, no. 3, pages 913-936, 2007.

[9] A. Bertozzi, J. Greer, S. Osher, and K. Vixie, Nonlinear regularizations of TV based PDEs for image process-ing , AMS Series of Contemporary Mathematics, vol. 371, pages 29-40, Gui-Qiang Chen, George Gasper,and Joseph Jerome eds, 2005.

[10] A. Bertozzi, C.-B. Schonlieb, Unconditionally stable schemes for higher order inpainting, in preparation.[11] C. Brito-Loeza, and K. Chen, Multigrid method for a modified curvature driven diffusion model for image

inpainting, Journal of Computational Mathematics, Vol.26, No.6, pp. 856–875, 2008.[12] M. Burger, and S. Osher, Convergence rates of convex variational regularization, Inverse Problems, Vol. 20,

Nr. 5, pp. 1411-1421(11), 2005.[13] M. Burger, E. Resmerita, L. He, Error Estimation for Bregman Iterations and Inverse Scale Space Methods

in Image Restoration, Computing, Vol. 81, pp. 109-135, 2007.[14] V. Caselles, J.-M. Morel, and C. Sbert, An axiomatic approach to image interpolation, IEEE Trans. Image

Processing, 7(3):376–386, 1998.[15] T.F. Chan, and S.H. Kang, Error Analysis for Image Inpainting, J. Math. Imaging Vis. 26, 1-2, 85-103, 2006.[16] T.F. Chan, S.H. Kang, and J. Shen, Euler’s elastica and curvature-based inpainting, SIAM J. Appl. Math.,

Vol. 63, Nr.2, pp.564–592, 2002.[17] T. F. Chan and J. Shen, Mathematical models for local non-texture inpaintings, SIAM J. Appl. Math.,

62(3):1019–1043, 2001.[18] T. F. Chan and J. Shen, Non-texture inpainting by curvature driven diffusions (CDD), J. Visual Comm.

Image Rep., 12(4):436–449, 2001.[19] T. F. Chan and J. Shen, Variational restoration of non-flat image features: models and algorithms, SIAM J.

Appl. Math., 61(4):1338–1361, 2001.[20] I. Daubechies, G. Teschke, and L. Vese, Iteratively solving linear inverse problems under general convex

constraints, Inverse Problems and Imaging (IPI), 1(1), pp.29-46, 2007.


[21] I.Ekeland, and R.Temam, Convex analysis and variational problems, Corrected Reprint Edition, SIAM,Philadelphia, 1999.

[22] S. Esedoglu and J. Shen, Digital inpainting based on the Mumford-Shah-Euler image model, Eur. J. Appl.Math., 13:4, pp. 353-370, 2002.

[23] L.C. Evans, Partial differential equations, Graduate Studies in Mathematics 19, American MathematicalSociety, Providence, RI, 1998.

[24] D. Eyre, An Unconditionally Stable One-Step Scheme for Gradient Systems, Jun. 1998, unpublished.[25] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhauser Boston 1984.[26] L. Lieu and L. Vese, Image restoration and decompostion via bounded total variation and negative Hilbert-

Sobolev spaces, Applied Mathematics & Optimization, Vol. 58, pp. 167-193, 2008.[27] O. M. Lysaker and Xue-C. Tai, Iterative image restoration combining total variation minimization and a

second-order functional, International Journal of Computer Vision, 66(1):5–18, 2006.[28] S. Masnou and J.-M. Morel, Level-lines based disocclusion, Proceedings of 5th IEEE Int’l Conf. on Image

Process., Chicago, 3:259–263, 1998.[29] G. Dal Maso, An introduction to Gamma-convergence, Birkhauser, Boston, 1993.[30] Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, The Fifteenth Dean

Jacqueline B. Lewis Memorial Lectures. American Mathematical Society, Boston, MA, USA, 2001.[31] L. Modica and S. Mortola, Il limite nella Γ-convergenza di una famiglia di funzionali ellittici, Boll. Unione

Mat. Ital., V. Ser., A 14, pp.526-529, 1977.[32] L. Modica, S. Mortola, Un esempio di Γ−-convergenza, Boll. Unione Mat. Ital., V. Ser., B 14, pp.285-299,

1977.[33] S. Osher, A. Sole, and L. Vese. Image decomposition and restoration using total variation minimization and

the H -1 norm, Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Vol. 1, Nr. 3,pp. 349-370, 2003.

[34] L. Rudin and S. Osher, Total variation based image restoration with free local constraints, Proc. 1st IEEEICIP, 1:31–35, 1994.

[35] L.I. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D,Vol. 60, Nr.1-4, pp.259-268, 1992.

[36] A. Tsai, Jr. A. Yezzi, and A. S. Willsky, Curve evolution implementation of the Mumford-Shah func-tional for image segmentation, denoising, interpolation and magnification, IEEE Trans. Image Process.,10(8):1169–1186, 2001.

[37] L. Vese, A study in the BV space of a denoising-deblurring variational problem, Applied Mathematics andOptimization, 44 (2), pp. 131-161, 2001.

[38] L. Vese, and S. Osher, Modeling textures with total variation minimization and oscillating patterns in imageprocessing, J. Sci. Comput., Vol. 19, Nr. 1-3, pp. 553-572, 2003.

Semantic Scholar€¦ · CAHN-HILLIARD INPAINTING AND A GENERALIZATION FOR GRAYVALUE IMAGES MARTIN BURGER∗, LIN HE†, AND CAROLA-BIBIANE SCHONLIEB¨ ‡ Abstract. The Cahn-Hilliard

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